Interval basic uncertain information and related aggregations in decision making

Abstract

The recently proposed basic uncertain information can directly present numerical uncertainties for given real values, but it cannot handle given interval values which themselves also have uncertainties. Against this background, this work proposes the concept of interval basic uncertain information which serves as a generalization of basic uncertain information and involves two types of uncertainties. We analyze some basic operations, weighted arithmetic mean and preference transformation for interval basic uncertain information. The Rule-based decisions and the comprehensive certainty of interval basic uncertain information are also discussed. An illustrative example of multi-source multi-criteria evaluation under interval basic uncertain information environment is presented.

Keywords

Aggregation operators basic uncertain information decision making interval basic uncertain information preference aggregation

1 Introduction

In numerous evaluation and decision making problems, the concerned data and information involve ever-increasing more uncertainties. The different types of uncertainties include interval information, fuzzy information, probability information, and some mathematical equivalences, extensions or variations of them including intuitionistic fuzzy information [1, 2], vague information [3], hesitance information [4], and so forth.

Handling different types of uncertainties usually may increase complexity and inconsistency in decision making, and therefore a general paradigm for uncertainty concept is convenient and flexible for practitioners to apply. Recently, basic uncertain information (BUI) [5, 6] has been proposed to serve as an uncertain paradigm which provides a uniform and quantified uncertainty measure in dealing with uncertain decision making. Based on BUI, many decision making, information fusion techniques and applications are soon developed by researchers [7 –13]. The study of BUI has also potential to be applied on many other areas such as in aggregation theory and computational intelligence [14 –22].

A BUI granule is expressed by a pair (a, c) ∈ [0, 1] ² in which a is the (main concerned) value while c is the certainty degree of a. In this paradigm, a large c indicates that a is known or obtained with high certainty, confidence or precision and vice versa. In addition, 1 - c is called the uncertain degree of a. When c = 1, a is considered to be the exact value; and c = 0 indicates that there is actually no accurate or effective information about the exact value of a. Due to the flexibility in practical meanings, BUI can be applied in different semantics. For example, when the predicted market share of a product is provided by 100a% from investigators, a group of m experts may believe it with different extents $(c_{j})_{j = 1}^{m} \in [0, 1]^{m}$ (which can be also regarded as fuzzy membership degrees in fuzzy sets theory). Taking the mean of them, we can obtain a certainty degree $c = (1 / m) \sum_{j = 1}^{m} c_{j}$ , and with the obtained BUI granule (a, c), decision makers may be able to better judge the situation and make some more reasonable decisions. Besides, in some situations the value a can be also explained as a fuzzy membership degree in fuzzy sets theory.

However, in practice decision makers may often be faced with some uncertainty information which itself is with interval form and still attached with a numerical certainty degree c. In actual, this situation is indeed reasonable and practical. The information decision makers require often is obtained in form of interval numbers and together with a numerical uncertain degree. For example, when an expert estimates the production rate of a factory in the next year, it may be an interval; when a collection of experts estimate, it may be several different values which after censoring can be also reported as an interval. Hence, a decision maker may still be only partly convinced with the reported interval (i.e., a numerical certainty degree c can be still assigned to it).

In general, an interval basic uncertain information (IBUI) granule can be expressed as the form ([a, b] , c). This work will present some detailed definitions, expressions, transformations, aggregations and applications of IBUI, as well as some basic applications in evaluation and decision making areas.

The remainder of this article is organized as follows. Section 2 provides with some basic concepts and related definitions. Section 3 formally proposes interval basic uncertain information with some related basic operations, aggregation and transformations. Section 4 discusses rule-based decisions and the comprehensive certainty of IBUI. In Section 5, we present some detailed decision procedures together with a numerical example for multi-source multi-criteria evaluation under IBUI environment. Section 6 concludes and remarks this work.

2 Preparations

This section reviews, rephrases or modifies some necessary definitions for this work.

Without loss of generality, the domain of discourse concerned in this work is closed interval within unit interval [a, b] ⊆ [0, 1], all of which are denoted by $I$ . In addition, [a, a] is often identified with real number a if there is no confusion arise. The set of all BUI granules (a, c) is denoted by $B$ .

For a finite vector of closed intervals, to avoid clutter we take the following convenient denotations when no confusion arising: $[a, b] = ([a_{i}, b_{i}])_{i = 1}^{n} \in I^{n}$ where $a = (a_{i})_{i = 1}^{n} \in [0, 1]^{n}$ and $b = (b_{i})_{i = 1}^{n} \in [0, 1]^{n}$ are real vectors. In a similar way, a BUI vector (of BUI granules) is defined by $(a, c) = (a_{i}, c_{i})_{i = 1}^{n} \in B^{n}$ where $a = (a_{i})_{i = 1}^{n} \in [0, 1]^{n}$ represents value vector while $c = (c_{i})_{i = 1}^{n} \in [0, 1]^{n}$ is the certainty (degree) vector corresponding to a.

We consider the interval lattice $(I, \leq_{Int})$ with the partial order ≤_Int such that [a₁, b₁] ≤ _Int [a₂, b₂] if and only if a₁ ≤ b₂ and b₁ ≤ b₂; we write [a₁, b₁] < _Int [a₂, b₂] if and only if [a₁, b₁] ≤ _Int [a₂, b₂] and [a₁, b₁] ≠ [a₂, b₂].

The weighted arithmetic mean and its modification for intervals are encapsulated below.

Definition 1.

A function WA_w : [0, 1] ⁿ → [0, 1] is called a weighted arithmetic mean (WAM) with (normalized) weight vector $w = (w_{i})_{i = 1}^{n}$ ( $\sum_{i = 1}^{n} w_{i} = 1$ and w_i ≥ 0 for all i ∈ N = {1, . . . , n}) if

${WA}_{w} (a) = \sum_{i = 1}^{n} w_{i} a_{i} .$ (1)

A function ${IWA}_{w} : I^{n} \to [0, 1]$ is called an interval weighted arithmetic mean (IWAM) with weight vector $w = (w_{i})_{i = 1}^{n}$ if

$\begin{matrix} {IWA}_{w} ([a, b]) & = \sum_{i = 1}^{n} w_{i} [a_{i}, b_{i}] \\ = [\sum_{i = 1}^{n} w_{i} a_{i}, \sum_{i = 1}^{n} w_{i} b_{i}] . \end{matrix}$ (2)

The definition of BUI weighted arithmetic mean and its related reasonability are presented in [5], and we formally rephrase the definition in the following definition.

Definition 2. The BUI weighted arithmetic mean (BUIWAM) with weight vector $w = (w_{i})_{i = 1}^{n}$ for BUI vector ${BWA}_{w} : B^{n} \to B$ is defined by

$\begin{matrix} {BWA}_{w} (a, c) & = \sum_{i = 1}^{n} w_{i} (a_{i}, c_{i}) \\ = (\sum_{i = 1}^{n} w_{i} a_{i}, \sum_{i = 1}^{n} w_{i} b_{i}) . \end{matrix}$ (3)

3 Interval basic uncertain information with some operations

Though we have briefly introduced the concept of IBUI in Introduction, its strict definition is formally stated as follows.

Definition 3. An interval basic uncertain information (IBUI) granules is denoted by the form ([a, b] , c) in which $[a, b] \in I$ is a closed interval value while c ∈ [0, 1] is the certainty degree of [a, b]. The uncertain degree of [a, b] is then given by 1 - c. In addition, the set of IBUI granules is denoted by $I B$ .

When [a, b] = [a, a], IBUI ([a, a] , c) then degenerates into BUI (a, c). Besides, in order to concisely write a vector of IBUI granules we adopt the following denotations as in the beginning of the preceding section. When there is no confusion arising we adopt the expression $([a, b], c) = ([a_{i}, b_{i}], c_{i})_{i = 1}^{n}$ . Accordingly, we will naturally take the following definition for interval BUI weighted arithmetic mean.

Definition 4. The interval BUI weighted arithmetic mean (IBUIWAM) with weight vector $w = (w_{i})_{i = 1}^{n}$ for IBUI vector ${IBWA}_{w} : {(I B)}^{n} \to I B$ is defined by

$\begin{matrix} {IBWA}_{w} ([a, b], c) = \sum_{i = 1}^{n} w_{i} ([a_{i}, b_{i}], c_{i}) \\ = ([\sum_{i = 1}^{n} w_{i} a_{i}, \sum_{i = 1}^{n} w_{i} b_{i}], \sum_{i = 1}^{n} w_{i} c_{i}) . \end{matrix}$ (4)

Recall that, for BUI granules (a, c), a usually used transformation for them to be converted into intervals are defined by [23] $T : B \to I$ such that for any $(a, c) \in B$ ,

$\begin{matrix} T (a, c) & = [a - (1 - c) a, a + (1 - c) (1 - a)] \\ = [ca, 1 - c + ca] \end{matrix}$ (5)

It is direct to check that if T (a, c) = [u, v] then u - v = 1 - c, the uncertainty degree of a.

Consider bi-polar optimism-pessimism preference degree α ∈ [0, 1] of which larger values represent more optimism and smaller values represent more pessimism. Often a bi-polar optimism-pessimism preference degree α ∈ [-1, 1] is also used in decision making, of which 1 represents the most optimism and -1 represents the most pessimism. Since the well-known Rademacher bijective mapping Rad : [0, 1] → [-1, 1] allows to map between [0, 1] and [-1, 1], then we may only consider the case of α ∈ [0, 1]. For the case where α ∈ [- ∞ , ∞], we may similarly use the so-called two point compactification to allow the mapping between [0, 1] and [- ∞ , ∞].

Subsequently, a preference mapping from intervals to real numbers $θ : I \times [0, 1] \to [0, 1]$ can be naturally obtained by

$θ ([a, b], α) = (1 - α) a + α b$ (6)

With the same principle, based on (5) a preference mapping from BUI granules to real numbers $φ : B \times [0, 1] \to [0, 1]$ can be obtained by

$φ ((a, c), β) = (1 - β) ca + β (1 - c + ca)$ (7)

By some interchangeable usages of the above (6) and (7) with preferences α and β, respectively, we next present some formulations to conveniently model the preference involved situation where both preferences α and β can be used in the IBUI environment.

Definition 5. The preference transformation (with preference α for interval and β for IBUI) $φ : I B \times [0, 1] \times [0, 1] \to [0, 1]$ is defined by

$φ (([a, b], c), α, β) = φ ((θ ([a, b], α), c), β)$ (8)

or by

$φ (([a, b], c), α, β) = θ ([φ ((a, c), β), φ ((b, c), β)], α)$ (9)

We next show the equivalence between (8) and (9).

Proposition 1. In Definition 5, φ ((θ ([a, b] , α) , c) , β) = θ ([φ ((a, c) , β) , φ ((b, c) , β)] , α).

Proof. $\begin{matrix} (i) φ ((θ ([a, b], α), c), β) \\ = φ (((1 - α) a + α b, c), β) \\ = (1 - β) c ((1 - α) a + α b) \\ + β (1 - c + c ((1 - α) a + α b)) \\ = (1 - β) (1 - α) ca + (1 - β) α cb + β (1 - c) \\ + β (1 - α) ca + β α cb \\ = (1 - α) ca + α cb + β (1 - c) . \end{matrix}$ $\begin{matrix} (ii) θ ([φ ((a, c), β), φ ((b, c), β)], α) \\ = θ ([(1 - β) ca + β (1 - c + ca), (1 - β) cb \\ + β (1 - c + cb)], α) \\ = (1 - α) ((1 - β) ca + β (1 - c + ca)) \\ + α ((1 - β) cb + β (1 - c + cb)) \\ = (1 - α) (1 - β) ca + (1 - α) β (1 - c + ca) \\ + α (1 - β) cb + α β (1 - c + cb) \\ = (1 - α) ca + (1 - α) β (1 - c) + α cb \\ + α β (1 - c) \\ = (1 - α) ca + α cb + β (1 - c) . \end{matrix}$ □

Remark. Note that when interval [a, b] is symmetrical (i.e., a ≤ 0.5 and a + b = 1), and both preferences are neutral (i.e., neither optimistic nor pessimistic), then such balanced situation coincides with the common intuition; that is, $\begin{matrix} φ (([a, 1 - a], c), 0.5, 0.5) \\ = (0.5) ca + (0.5) c (1 - a) + (0.5) (1 - c) = 0.5 . \end{matrix}$

We may also have only one preference (i.e., either with preference α for interval or with β for BUI) and define the following separated definitions, which presents more flexibility for decision makers in practices.

Definition 6. The preference transformation with preference α for interval $φ_{I} : I B \times [0, 1] \to B$ is defined by

$φ_{I} (([a, b], c), α) = (θ ([a, b], α), c)$ (10)

Similarly, the preference transformation with preference β for BUI $φ_{B} : I B \times [0, 1] \to I$ is defined by

$φ_{B} (([a, b], c), β) = [φ ((a, c), β), φ ((b, c), β)]$ (11)

4 Rule-based decisions with IBUI and the comprehensive certainty

Rule-based decisions are often used in decision making and some other related areas [24 –26]. The basic rule-based decisions involve If-Then structure. This structure can have numerous flexible extension rules which are particularly suitable to the defining of decision rules and procedures in a wide range of BUI based decision making problems. In IBUI decision environment, we may designs some reasonable rules as the intelligent decision aid for practitioners to make evaluations and decisions.

First, we present some rules as decision thresholds to judge whether some values can be qualified. Given any IBUI granule $([a, b], c) \in I B$ , we may design the following rules by a general principle that “value” and “certainty” simultaneously attaining some thresholds is an reasonable indication for the related IBUI granule to become qualified.

If [a, b] _nleqInt [u, v] and c ≥ y, then ([a, b] , c) is qualified.

If a ≥ u and c ≥ y, then ([a, b] , c) is qualified.

If b ≥ u and c ≥ y, then ([a, b] , c) is qualified.

If 0.5 (a + b) ≥ u and ac ≥ y, then ([a, b] , c) is qualified.

If b - a ≤ x and c ≥ y, then ([a, b] , c) is qualified.

If 0.5 (a + b) ≥ u and (1 - b + a) ∘ c ≥ y, then ([a, b] , c) is qualified. “∘” is a given semi-copula, i.e., it is non-decreasing binary aggregation operator [17] and satisfies x ∘ 1 =1 ∘ x = x for any x ∈ [0, 1].

We can also take decisions based on categorical variable set $V^{(k)} = {v_{i}}_{i = 1}^{k}$ such as V⁽⁴⁾ = {v₀, v₁, v₂, v₃}=/{0 unknown, 1 not qualified, 2 qualified, 3 excellent/}. For example,

If a ≥ u = 0.7 and c ≥ y = 0.7, then ([a, b] , c) is “3 excellent”;

else if b < v = 0.5 and c ≥ y = 0.8, then ([a, b] , c) is “1 not qualified”;

else if 0.5 (a + b) ≥ u = 0.6 and c ≥ y = 0.5, then ([a, b] , c) is “2 qualified”;

else ([a, b] , c) is “0 unknown”.

Second, for two IBUI granules $([a_{1}, b_{1}], c_{1}), ([a_{2}, b_{2}], c_{2}) \in I B$ , we can design some decision rules to compare them and then help decision makers to choose a better one.

If [a₁, b₁] ≤ _Int [a₂, b₂] and min {c₁, c₂} ≥ y = 0.5, then [a₂, b₂] is better than [a₁, b₁];

else if a₁ + b₁ ≤ a₂ + b₂ and min {c₁, c₂} ≥ y = 0.7, then [a₂, b₂] is better than [a₁, b₁];

else the two IBUI granules cannot be compared with current information.

The comparison can involve bi-polar optimism-pessimism preference for both IBUI granules. For example,

If φ (([a₁, b₁] , c₁) , α, β) < φ (([a₂, b₂] , c₂) , α, β), then [a₂, b₂] is better than [a₁, b₁].

We next discuss the inherent uncertainties of IBUI granules. Given any IBUI granule ([a, b] , c), it involves two types of uncertainties rather than only one type as in BUI granule (a, c). This is because interval [a, b] also has its innate uncertainty which usually can be expressed by the quantity b - a (or, its innate certainty is 1 - b + a). Therefore, in order to compare the quantities of uncertainty of two or more IBUI granules, we should simultaneously consider the two types of uncertainties. One usually considered method to handle such problem is to use binary aggregation operators. For example, we may use a certain semi-copula ∘ or t-norm t : [0, 1] ⁿ → [0, 1] to aggregate the two parts of certainty in one IBUI granule, i.e., 1 - b + a and c, to obtain a comprehensive certainty (1 - b + a) ∘ c or t (1 - b + a, c).

Another plausible method to derive certainty/uncertainty extent is by considering the Lebesgue measure of all the points that can or cannot be the exact concerned value. For example, for an interval [a, b] which contains the exact concerned value, it can be concluded that all the points in [0, 1] ∖ [a, b] = [0, a) ∪ (b, 1] are impossible to be the exact value. Hence, for IBUI granule ([a, b] , c), since applying (5) to transform a BUI granule into interval is reasonable, then using (5) twice to obtain two corresponding intervals T (a, c) and T (b, c), the points belonging to T (a, c) ∪ T (b, c) = [ca, 1 - c + ca] ∪ [cb, 1 - c + cb] = [ca, 1 - c + cb] are possible to be the exact concerned value. Therefore, we may define the certainty of ([a, b] , c) to be 1 - (1 - c + cb) + ca = c (1 - b + a), which is plausible and conforms to intuition in both reasonability and mathematical consistency.

Decision makers can order or prioritize a vector of IBUI granules according to the above discussed certainty measurement via some preference vectors, e.g., directly from some given OWA weight vectors [27] or generated by using RIM quantifiers [28]. For example, for an IBUI vector ([a, b] , c) = ([a_i, b_i] , c_i) we can choose an OWA weight vector $w = (w_{i})_{i = 1}^{n}$ such that w₁ ≥ w₂ ≥ ⋯ ≥ w_n (which implies orness (w) ≥0.5 indicating a preference over the IBUI granules ([a_i, b_i] , c_i) with smaller uncertainties).

An alternative method to generate a similar preference vector is to directly derive from the IBUI vector itself. For example, we may form a weight vector $w = (w_{i})_{i = 1}^{n}$ such that $w_{i} = (1 - b_{i} + a_{i}) \circ c_{i} / \sum_{k = 1}^{n} (1 - b_{k} + a_{k}) \circ c_{k}$ . However, when (1 - b_i + a_i) ∘ c_i = 0 for all i ∈ {1, . . . , n}, this method does not work and hence we may modify the defect with an addition of positive quantity s > 0 to the denominator so that $w_{i} = ((1 - b_{i} + a_{i}) \circ c_{i} + s) / (ns + \sum_{k = 1}^{n} (1 - b_{k} + a_{k}) \circ c_{k})$ .

With the obtained preference vector w by either of the two methods discussed above, an IBUIWAM ${IBWA}_{w} : {(I B)}^{n} \to I B$ can be applied using (4) to yield a final aggregated IBUI granule under such preference. Such preference intervened weight allocation and aggregation can have wide applications in real decision making practices.

5 IBUI in multi-source multi-criteria evaluation

In this section, we show some decision making procedures of IBUI with its operations in multi-source multi-criteria (MSMC) evaluation. Though with sufficient reasonability, the following procedures still have prescriptive sense and hence may be modified in practices according to different decisional backgrounds and situations.

The whole IBUI-MSMC evaluation procedures include several steps and is listed sequentially as follows.

Step 1 Determine m information sources ${S_{j}}_{j = 1}^{m}$ from which individual evaluations can be obtained. Decide an original weight vector $v = (v_{j})_{j = 1}^{m}$ to ${S_{j}}_{j = 1}^{m}$ , with v_j representing the relative importance of information source S_j.

Step 2 Determine n criteria ${C_{i}}_{i = 1}^{n}$ (it in general may include 1–3 layers depending on detail decisional scenarios) for performing comprehensive evaluation. Decide a weight vector $w = (w_{i})_{i = 1}^{n}$ to ${C_{i}}_{i = 1}^{n}$ , with w_i representing the relative importance of criterion C_i.

Step 3 For each information source and each criterion, obtain one individual evaluation which is IBUI granule $([a_{ji}, b_{ji}], c_{ji}) \in I B$ (j ∈ {1, . . . , m} , i ∈ {1, . . . , n}); that is, obtain IBUI vectors ([a_j, b_j] , c_j) (j ∈ {1, . . . , m}).

Step 4 For each j ∈ {1, . . . , m}, perform IBUIWAM IBWA_w with weight vector w and input ([a_j, b_j] , c_j): $\begin{matrix} {IBWA}_{w} ([a_{j}, b_{j}], c_{j}) = \sum_{i = 1}^{n} w_{i} ([a_{ji}, b_{ji}], c_{ji}) \\ = ([\sum_{i = 1}^{n} w_{i} a_{ji}, \sum_{i = 1}^{n} w_{i} b_{ji}], \sum_{i = 1}^{n} w_{i} c_{ji}), \end{matrix}$ and obtain an IBUI vector $([u, v], d) = ([u_{j}, v_{j}], d_{j})_{j = 1}^{m} = {({IBWA}_{w} ([a_{j}, b_{j}], c_{j}))}_{j = 1}^{m}$ .

Step 5 If for all information sources S_j (j ∈ {1, . . . , m}), we have $h_{j}^{'} = (1 - v_{j} + u_{j}) \circ d_{j} = 0$ , then this shows that all the information sources are invalid and the information in need should be recollected and stop the evaluation procedures. If not so, then form a weight vector $h = (h_{j})_{j = 1}^{m} = (h_{j}^{'} / \sum_{k = 1}^{m} h_{k}^{'})$ . It is reasonable that we make a large weight h_j correspond to a more convincible information source S_j.

Step 6 Since the original weight vector $v = (v_{j})_{j = 1}^{m}$ assigned to ${S_{j}}_{j = 1}^{m}$ still should be considered, then we may have a parameter λ ∈ [0, 1] (indicating the extent by which we prefer using h) and use the combination form q = λh + (1 - λ) v as a new modified weight vector.

Step 7 With the obtained IBUI vector ([u, v] , d) and weight vector q, perform IBUIWAM IBWA_q: $\begin{matrix} {IBWA}_{q} ([u, v], d) = \sum_{j = 1}^{m} q_{j} ([u_{j}, v_{j}], d_{j}) \\ = \sum_{j = 1}^{m} ([q_{j} u_{j}, q_{j} v_{j}], q_{j} d_{j}) . \end{matrix}$

Step 8 With the obtained final IBUI ([a, b] , c) = IBWA_q ([u, v] , d), and the preset decision rules and evaluation thresholds, we may take some corresponding evaluation.

The comprehensive evaluation on university teachers is important and with some difficulties. This is because university teachers in general are also scholars, and therefore acceptable academic evaluation should also be made for the teachers. Since many evaluation criteria are based on subjective evaluation, then it is suitable to invite several different experts to make individual subjective evaluations. IBUI can be quite suitable to be applied in subjectivity involved evaluations for its flexibility and simplicity in modeling, measuring, expressing and aggregating quantifiable uncertainties.

As an illustrative case for the foregoing introduced IBUI-MSMC evaluation procedures, we propose to use the following single layer criteria system for comprehensively evaluating university teachers. The set of criteria includes the following four aspects.

C1: Teaching attitude and method

C2: Teaching effect and result

C3: Academic attitude and morality

C4: Scientific study and research

The other steps of the detailed numerical example are illustrated as follows.

Step 1 Find 3 information sources ${S_{j}}_{j = 1}^{3}$ which represent three different education evaluation experts, from which individual evaluations for one teacher can be obtained. Decide an original weight vector $v = (v_{j})_{j = 1}^{3} = (0.4, 0.3, 0.3)$ to ${S_{j}}_{j = 1}^{3}$ , with v_j representing the relative importance of expert S_j.

Step 2 Set 4 criteria ${C_{i}}_{i = 1}^{4}$ for performing comprehensive evaluation. Decide a weight vector $w = (w_{i})_{i = 1}^{4} = (0.2, 0.1, 0.4, 0.3)$ to ${C_{i}}_{i = 1}^{4}$ , with w_i representing the relative importance of criterion C_i.

Step 3 For each expert and each criterion, obtain one individual evaluation which is IBUI granule $([a_{ji}, b_{ji}], c_{ji}) \in I B$ (j ∈ {1, 2, 3} , i ∈ {1, 2, 3, 4}) and obtain IBUI vectors ([a_j, b_j] , c_j) (j ∈ {1, 2, 3}) as listed in the following: $\begin{matrix} \begin{matrix} ([a_{11}, b_{11}], c_{11}) = ([0.6, 0.9], 0.7); \\ ([a_{12}, b_{12}], c_{12}) = ([0.5, 0.7], 0.9); \\ ([a_{13}, b_{13}], c_{13}) = ([0.6, 1], 1); \\ ([a_{14}, b_{14}], c_{14}) = ([0.2, 0.8], 0.4) . \\ ([a_{21}, b_{21}], c_{21}) = ([0.5, 0.7], 0.7); \\ ([a_{22}, b_{22}], c_{22}) = ([0.8, 0.8], 1); \\ ([a_{23}, b_{23}], c_{23}) = ([1, 1], 1); \\ ([a_{24}, b_{24}], c_{24}) = ([0.5, 0.9], 0.7) . \\ ([a_{31}, b_{31}], c_{31}) = ([0.2, 0.6], 0.9); \\ ([a_{32}, b_{32}], c_{32}) = ([0.5, 0.5], 0.5); \\ ([a_{33}, b_{33}], c_{33}) = ([0.8, 1], 0.6); \\ ([a_{34}, b_{34}], c_{34}) = ([0.5, 0.8], 0.2) . \end{matrix} \end{matrix}$

Step 4 For each j ∈ {1, 2, 3}, perform IBUIWAM IBWA_w with weight vector w and input ([a_j, b_j] , c_j): $\begin{matrix} \begin{matrix} {IBWA}_{w} ([a_{1}, b_{1}], c_{1}) \\ = ([\sum_{i = 1}^{4} w_{i} a_{1 i}, \sum_{i = 1}^{4} w_{i} b_{1 i}], \sum_{i = 1}^{4} w_{i} c_{1 i}) \\ = ([0.12, 0.18], 0.14) + ([0.05, 0.07], 0.09) \\ + ([0.24, 0.4], 0.4) + ([0.06, 0.24], 0.12) \\ = ([0.47, 0.89], 0.75); \\ {IBWA}_{w} ([a_{2}, b_{2}], c_{2}) \\ = ([\sum_{i = 1}^{4} w_{i} a_{2 i}, \sum_{i = 1}^{4} w_{i} b_{2 i}], \sum_{i = 1}^{4} w_{i} c_{2 i}) \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} = ([0.1, 0.14], 0.14) + ([0.08, 0.08], 0.1) \\ + ([0.4, 0.4], 0.4) + ([0.15, 0.27], 0.21) \\ = ([0.73, 0.89], 0.85); \\ {IBWA}_{w} ([a_{3}, b_{3}], c_{3}) \\ = ([\sum_{i = 1}^{4} w_{i} a_{3 i}, \sum_{i = 1}^{4} w_{i} b_{3 i}], \sum_{i = 1}^{4} w_{i} c_{3 i}) \\ = ([0.04, 0.12], 0.18) + ([0.05, 0.05], 0.05) \\ + ([0.32, 0.4], 0.24) + ([0.15, 0.24], 0.06) \\ = ([0.56, 0.81], 0.53) . \end{matrix} \end{matrix}$ and obtain an IBUI vector

$\begin{matrix} \begin{matrix} ([u, v], d) = ([u_{j}, v_{j}], d_{j})_{j = 1}^{3} = (([0.47, 0.89], 0.75), \\ ([0.73, 0.89], 0.85), ([0.56, 0.81], 0.53)) . \end{matrix} \end{matrix}$

Step 5 For all information sources S_j (j ∈ {1, 2, 3}), we adopt product “·” as the semi-copula and observe $h_{j}^{'} = (1 - v_{j} + u_{j}) \cdot d_{j} > 0$ , and form a weight vector $\begin{matrix} \begin{matrix} h = (h_{j})_{j = 1}^{3} = (h_{j}^{'} / \sum_{k = 1}^{3} h_{k}^{'}) = (0.435, 0.714, \\ 0.3975) / 1.5465 = (0.281, 0.462, 0.257) . \end{matrix} \end{matrix}$

Step 6 We take a parameter λ = 0.5 (indicating we equally consider v and h) and use the combination form q = λh + (1 - λ) v = 0.5h + 0.5v = (0.3405, 0.381, 0.2785) as the new modified weight vector.

Step 7 With the obtained IBUI vector ([u, v] , d) and weight vector q, perform IBUIWAM IBWA_q: $\begin{matrix} {IBWA}_{q} ([u, v], d) = ([\sum_{j = 1}^{3} q_{j} u_{j}, \sum_{j = 1}^{3} q_{j} v_{j}], \\ \sum_{j = 1}^{3} q_{j} d_{j}) = ([0.594, 0.868], 0.727) . \end{matrix}$

Step 8 Suppose we preset a threshold to be an IBUI ([0.5, 0.8] , 0.65), and predetermine that if the comprehensive resulting IBUI granule ([a, b] , c) satisfies [0.5, 0.8] ≤ _Int [a, b] and 0.65 ≤ c, then the teacher is qualified. Clearly, with the obtained resulting IBUI ([0.594, 0.868] , 0.727) we make the evaluation that the teacher is qualified. In practice, university managements can make and flexibly adjust some suitable thresholds for comparison and obtain different evaluation results according to varying and complex situations.

6 Conclusions

The concept of BUI has proved very useful and flexible in numerous applications and attracted ever-increasingly more attentions and interests from researchers and practitioners. The article further extended BUI into IBUI, and IBUI granule ([a, b] , c) is a good generalization of BUI granule (a, c).

Some operations, aggregations and transformations of IBUI were also analyzed. The weighted arithmetic mean for IBUI inherits the corresponding definitions of BUI and still proves reasonable. The preference transformation with preference α for interval and β for IBUI was elaborately analyzed and some of its flexible variations are also discussed. The results can help managements make decision and evaluation with bi-polar optimism-pessimism preferences involved.

We showed IBUI can be well applied in rule-based decisions making and proposed several different IBUI based decision rules which can be further extended or modified according to different decisional scenarios. Some methods to derive comprehensive certainty of IBUI and their reasonability were also discussed.

A set of decision procedures and a numerical example for multi-source multi-criteria evaluation under IBUI environment were also proposed, indicating the good potential of IBUI to be applied in more practices. Some parts of the results in this work also have theoretical values in aggregation theory and information fusion.

As intervals are themselves with uncertainties [29], in decision making they will significantly affect the decision processes and results. In mathematics, BUI granules actually can also serve as a perfect generalization of intervals and thus IBUI can be a generalization of twin intervals, which will be further strictly analyzed in our later study. Moreover, as we analyzed in Section 4, in practice BUI and IBUI can provide more decision making rules and methods than interval and twin intervals, and they may derive more practical, effective and flexible decision making methods and theories from researchers in a wide range of areas.

Footnotes

Acknowledgment

This work is supported by Grant: VEGA 1/0006/19 and Grant: APVV-0052-18.

References

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets Syst 20(1) (1986), 87–96.

Atanassov

K.T.

and Gargov

, Interval valued intuitionistic fuzzy sets, Fuzzy Sets Syst 31(3) (1989), 343–349.

Gau

and Buehrer

, Vague sets, IEEE Trans. Syst. Man Cybern 23(2) (1993), 610–614.

Torra

, Hesitant fuzzy sets, Int. J. Intell. Syst. 25 (2010), 529–539.

Jin

, Kalina

, Mesiar

and Borkotokey

, Certainty Aggregation and the Certainty Fuzzy Measures, Int. J. Intell. Syst. 33(4) (2018), 759–770.

Mesiar

, Borkotokey

, Jin

and Kalina

, Aggregation under uncertainty, IEEE Trans. Fuzzy Syst 26(4) (2018), 2475–2478.

Tao

, Shao

, Liu

, Zhou

and Chen

, Basic uncertain information soft set and its application to multi-criteria group decision making, Engineering Applications of Artificial Intelligence 95 (2020), 103871.

Chen

Z.S.

, Martinez

, Chang

J.P.

, Wang

X.J.

, Xiong

S.H.

and Chin

K.S.

, Sustainable building material selection: A QFD- and ELECTRE III-embedded hybrid MCGDM approach with consensus building, Engineering Applications of Artificial Intelligence 85 (2019), 783–807.

Chen

Z.S.

, Wang

X.J.

, Chin

K.S.

, Tsui

K.L.

and Martinez

, Individual Semantics Building for HFLTS Possibility Distribution With Applications in Domain-Specific Collaborative Decision Making, IEEE Access 6 (2018), 78803–78828.

10.

Borkotokey

, Mesiar

, Li

, Kouchakinejad

and Siposova

, Event-based transformations of capacities and invariantness, Soft Computing 22 (2018), 6291–6297.

11.

Tao

, Liu

, Zhou

and Chen

, Rank aggregation based multi-attribute decision making with hybrid Z-information and its application, Journal of Intelligent and Fuzzy Systems 37(3) (2019), 1–9.

12.

Liu

and Xiao

, An interval-valued exceedance method in MCDM with uncertain satisfactions, International Journal of Intelligent Systems 34(10) (2019), 2676–2691.

13.

Chen

Z.S.

, Martinez

, Chin

K.-S.

and Tsui

K.-L.

, Two-stage Aggregation Paradigm for HFLTS Possibility Distributions: A Hierarchical Clustering Perspective, Expert Systems with Applications 104(15) (2018), 43–66.

14.

Chen

Z.S.

, Yu

, Chin

and Martínez

, An enhanced ordered weighted averaging operators generation algorithm with applications for multicriteria decision making, Applied Mathematical Modelling 71 (2019), 467–490.

15.

Chen

Z.S.

, Zhang

, Rodríguez

R.M.

, Pedrycz

and Martínez

, Expertise-based bid evaluation for construction-contractor selection with generalized comparative linguistic ELECTRE III, Automation in Construction 125 (2021), 103578.

16.

Chen

Z.S.

, Liu

X.L.

, Chin

K.S.

, Pedrycz

, Tsui

K.L.

and Skibniewski

M.J.

, Online-review analysis based large-scale group decision-making for determining passenger demands and evaluating passenger satisfaction: Case study of high-speed rail system in China, Information Fusion 69 (2021), 22–39.

17.

Grabisch

, Marichal

J.L.

, Mesiar

and Pap

, Aggregation Functions, Cambridge University Press (2009), Cambridge, ISBN:1107013429.

18.

Klement

E.P.

, Mesiar

and Pap

, Triangular norms, Springer-Verlag, Kluwer, Dordrecht, (2000).

19.

Muhiuddin

, Al-Kadi

and Balamurugan

, Anti-Intuitionistic Fuzzy Soft a-Ideals Applied to BCI-Algebras, Axioms 9(3) (2020), 79. doi: 10.3390/axioms9030079.

20.

Tiwari

, Generalized Entropy and Similarity Measure for Interval-Valued Intuitionistic Fuzzy Sets With Application in Decision Making, International Journal of Fuzzy System Applications (IJFSA) 10(1) (2021), 64–93. doi: 10.4018/IJFSA.2021010104.

21.

Boczek

, Hovana

and Hutnik

, General form of Chebyshev type inequality for generalized Sugeno integral, International Journal of Approximate Reasoning 115 (2019), 1–12.

22.

Boczek

, Hovana

, Hutník

and Kaluszka

, New monotone measure-based integrals inspired by scientific impact problem, European Journal of Operational Research 290 (2021), 346–357.

23.

Bustince

and Burillo

, Correlation of interval-valued intuitionistic fuzzy sets, Fuzzy Sets Syst 74(2) (1995), 237–244.

24.

Zadeh

L.A.

, Outline of a New Approach to Analysis of Complex Systems and Decision Processes, IEEE Transactions on Systems, Man and Cybernetics 1 (1973), 28–44.

25.

Takagi

and Sugeno

, Fuzzy identification of systems and its application to modeling and control, IEEE Trans. on Syst., Man & Cybernetics 15 (1985), 116–132.

26.

Pedrycz

, Fuzzy relational equations with generalized connectives and their applications, Fuzzy Sets and Systems 10(1–3) (1983), 185–201.

27.

Yager

R.R.

, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Syst. Man Cybern. 18(1) (1988), 183–190.

28.

Yager

R.R.

, Quantifier guided aggregation using OWA operators, Int J Intell Syst 11 (1996), 49–73.

29.

Kreinovich

, Decision making under interval (and more general) uncertainty: monetary vs. utility approaches, Journal of Computational Technologies 22(2) (2017), 37–49.